How accurate would this analogy be between derivatives and intermediate fields? In Galois theory, you have a field extension $L : M$ and an intermediate field $K$, and $L : M$ tells you something about $L : K$. Next, you have a function, a first order derivative, and a second order derivative. The first order derivative kind of reminds me of an intermediate field. Can anyone comment on this analogy and tell me if there is some sort of deep impetus behind it? 
 A: No, there is no deep connection here, nor even any connection besides both being "intermediate" in some very general sense.
(I realize this is a short and perhaps seemingly unhelpful response, but I don't know what more there is to say.  If someone came up to you and asked whether there was some kind of deep connection between chairs and ladybugs because both have legs, how would you respond?  If you elaborate on exactly what sort of connection you might hope to find, I may be able to say more.)
A: The analogy is not very accurate. Let us consider a small example, the field extension$$L = \mathbb{Q}(\sqrt{2}, i)/\mathbb{Q}.$$This is a degree $4$ Galois extension with Galois group $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. There are three intermediate fields between $L$ and $\mathbb{Q}$:$$K_1 = \mathbb{Q}(\sqrt{2}),\text{ }K_2 = \mathbb{Q}(i),\text{ and }K_3 = \mathbb{Q}(\sqrt{-2}).$$Which of these corresponds to the first derivative of a function? Which one corresponds to the second derivative? Third derivative?
The first derivative of a differentiable function completely determines the function up to a constant. How does an intermediate field extension "determine" a larger field extension?
I am failing to see a deep connection here. Also, here is a related question. It is well-known that there is a $1$-$1$ correspondence between intermediate fields of the extension $L/K$ and subgroups of the Galois group $\text{Gal}(L/K)$. Furthermore, this correspondence identifies Galois extensions of $K$ contained in $L$ with normal subgroups of $\text{Gal}(L/K)$.
So if there was an analogy between intermediate fields and derivatives, then by extension, we would have an analogy between derivatives and subgroups of a Galois group. If your analogy really does make sense, then you should be able to explain it in these terms, and I am not sure how one would do that.
This is also a helpful thing to do in math all the time. Let us say you are proving something about intermediate fields. It is useful then to try and translate that statement into a statement about subgroups of the Galois group. Often this can provide more insight to the theorem and help make connections with other theorems.
